MCMC methods for continuous time multi-state models and high dimensional copula models

In this Thesis we propose Markov chain Monte Carlo (MCMC) methods for several classes of models. We consider both parametric and nonparametric Bayesian approaches proposing either alternatives in computation to already existent methods or new computational tools. In particular, we consider continuous time multi-state models (CTMSM), that is a class of stochastic processes useful for modelling several phenomena evolving continuously in time, with a finite number of states. Inference for these models is straightforward if the processes are fully observed, while it presents some computational difficulties if the processes are discretely observed and there is no additional information about the state transitions. In particular, in the semi-Markov models case the likelihood function is not available in closed form and approximation techniques are required. In the first Chapter we provide a uniformization based algorithm for simulating continuous time semi-Markov trajectories between discretely observed points and propose a Metropolis within Gibbs algorithm in order to sample from the posterior distributions of the parameters of that class of processes. As it will be shown, our method generalizes the Markov case. In the second Chapter we present a novel Bayesian nonparametric approach for inference on CTMSM. We propose a Dirichlet Process Mixture with continuous time Markov multi-state kernels, providing a Gibbs sampler which exploit the conjugacy between the Markov CTMSM density and the chosen base measure. The method, that is applicable with fully observed and discretely observed data, represents a flexible solution which avoid parametric assumptions on the process and allows to get density estimation and clustering. In the last Chapter we focus on copulas, a class of models for dependence between random variables. The copula approach allows for the construction of joint distributions as product of marginals and copula function. In particular, we focus on the modelling of the dependence between more than two random variables. In that case, assuming a multidimensional copula model for the multivariate data implies that paired data dependencies are assumed to belong to the same parametric family. This constraint makes this class of models not very flexible. A proposed solution to this problem is the vine copula constructions, which allows us to rewrite the multivariate copula as product of pair-copulas which may belong to different copula families. Another solution may be the nonparametric approach. We present two Bayesian nonparametric methods for inference on copulas in high dimensions. The first proposal is an alternative to an already existent method for high dimensional copulas. The second method is a novel Dirichlet Process Mixture of conditional multivariate copulas, which accounts for covariates on the dependence between the considered variables. Applications with both simulated and real data are provided in the last section of the first and the second Chapters, while in the last Chapter there are only application with simulated data.